Greenhouse Law formula is expressed below, where C is ______ and F is

______:

ΔF = α Ln(C/C

0

)

a.

a.

carbon dioxide (CO

2

) concentration, the radiative forcing

b.

carbon monoxide (CO) concentration, the frictional forcing

c.

the centrifugal forcing, the solar forcing

d.

carbon dioxide (CO

2

), the magnetic forcing

e.

a.

both b and c

5 points

·

·

Question 2

(short

–

answer question

)

·

·

At the beginning of the industrial revolution in 1850, the CO

2

concentration

was 280 ppm. Today, it is 416 ppm. 1) If

CO

2

concentration reaches 500

ppm, how much extra radiativ

e forcing is the Earth’s surface receiving,

relative to 1850? 2) What is the equivalent temperature change?

·

ΔF (Wm

–

2

) = α ln(C/C

0

),

ΔT(K) = λ*ΔF

·

α = 5.35, λ = 0.8 per (Wm

–

2

)

Question 3

Typically, the time of day when relative humidity is highest is ________.

a.

at dawn

b.

in the evening

c.

in the late afternoon

d.

at midnight

e.

at sunset

5 points

Greenhouse Law formula is expressed below, where C is ______ and F is

______:

ΔF = α Ln(C/C0)

a.

a.

carbon dioxide (CO2) concentration, the radiative forcing

b. carbon monoxide (CO) concentration, the frictional forcing

c.

the centrifugal forcing, the solar forcing

d. carbon dioxide (CO2), the magnetic forcing

e.

a. both b and c

5 points



 Question 2 (short-answer question)



 At the beginning of the industrial revolution in 1850, the CO2 concentration

was 280 ppm. Today, it is 416 ppm. 1) If CO2 concentration reaches 500

ppm, how much extra radiative forcing is the Earth’s surface receiving,

relative to 1850? 2) What is the equivalent temperature change?

 ΔF (Wm

-2

) = α ln(C/C0), ΔT(K) = λ*ΔF

 α = 5.35, λ = 0.8 per (Wm

-2

)

Question 3

Typically, the time of day when relative humidity is highest is ________.

a. at dawn

b.

in the evening

c. in the late afternoon

d. at midnight

e.

at sunset

5 points

�if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression� (1896)

ΔF (Wm-2) = α ln(C/C0) α = 5.35

ΔT(K) = λ*ΔF

Svante Arrhenius Climate Sensitivity: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.

Let’s talk about climate sensitivity with a little more specificity in terms of CO2 emissions. Once again, as shared in previous lecture slides, this slide shows the “greenhouse law” formulated by Svante Arrhenius.

CO2 carbon dioxide CH4 methane

N2O nitrous oxide

It is important to note that, although we are focusing on CO2 to examine climate sensitivity here, we all know that there are other molecules that can absorb long wave back radiation: water molecule (H2O), methane (CH4), nitrous oxide (N2O), etc.

What is a logarithm? • logbX “the log to base b of X” is the power that b

must be raised to get X

• So, what is log10100? (= 2)

• Logs with the base e (LogeX) are written in Ln X (natural logarithm)

• e=2.71828…. (mathematical constant)

• Definition:

Arrhenius’s greenhouse law tells us that the relationship between climate

sensitivity and carbon dioxide (CO2) concentration is a natural logarithm.

ΔF (Wm-2) = α ln(C/C0)

This means that… (see next slide)

Radiative ‘forcing’ of CO2 is logarithmic!

Increasing CO2 concentration à

In cr

ea si

ng ra

di at

iv e

fo rc

in g

in W

m 2

Radiative forcing, defined as the difference between radiant energy received by the Earth and energy re-radiated to space logarithmically increases with increasing CO2 concentration. More importantly, the increase in radiative forcing slows down as CO2 concentration increases.

�if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression� (1896)

ΔF (Wm-2) = α ln(C/C0) α = 5.35

ΔT(K) = λ*ΔF

Svante Arrhenius Climate Sensitivity: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.

At the beginning of the industrial revolution, CO2 concentration was 280 ppmv (parts per million by volume)

As of May 2013, CO2 concentration was 398 ppmv.

How much extra radiative forcing is the Earth’s surface getting in 2013, relative to 1850?

�if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression� (1896)

Radiative forcing since 1850 AD

5.35Wm−2 •ln 398ppm 280ppm "

# $

%

& '

=1.9Wm−2

We must multiply this times a “sensitivity factor” λ to calculate temperature change

λ is usually assumed to be about 0.8 per (Wm-2). So, 0.8 * 1.9 = ~1.5K (kelvin).

Planetary Energy Budget

Through this week’s lectures, we will learn about the planetary energy budget and use this knowledge to understand a climate simulation model. This model will be key to developing a better understanding about our future.

As we learned in previous lectures, this figure demonstrates the incoming and outgoing energy budget.

Stefan-Boltzman Law • Energy intensity radiated by a perfect radiator

(blackbody) is: E=sT4 – Where E is a combination of intensity and

emissivity; represents the total rate of energy emission from the object at all frequencies

– Where sigma (s) is a fundamental constant of physics that never changes, also known as the Stefan-Boltzmann constant: s = 5.67 x 10-8 W m-2 K-4

– Where T is the temperature in Kelvins

We have learned that infrared light emission is associated with a blackbody, and its spectrum depends on the temperature of the object. There is an equation that allows us to identify how quickly energy is radiated from a blackbody object. This is called the Stefan-Boltzmann equation.

T, the temperature in Kelvins, is expressed as the superscript 4, which is an exponent. The Kelvin temperature scale begins with 0 K at which point atoms vibrate/move as little as possible; a temperature called absolute zero. Please note that there are no negative temperatures on the Kelvin scale.

So what does this Stefan-Boltzman equation tells us? — A hot object emits much more light than a cold object!

Simple Climate Model

• Energy coming in from Sun must Equal Energy radiating back into space: in=out.

S/4(1-Albedo) = radiation out to space S/4(1-Albedo) = sT4

T = (S/4(1-Albedo)/s)1/4

In climate science, “models” are used in two different ways. One way is to make forecasts. For this purpose, a model should be as realistic as possible and should capture or include all processes that might be relevant in nature – typically, mathematical models are implemented on a computer. Once such a model has been constructed, a climate scientist can perform “what-if” experiments on it that could never be done in the real world, to determine how sensitive the climate would be to changes in the brightness of the sun or properties of the atmosphere, for instance.

We can also create a simple climate model by ourselves. Here, let’s try to construct a simple model called a “Layer Model”. This simple model is not intended to make detailed forecasts of our future climate, rather, it is used to better understand the workings of the real climate system.

In a Layer Model, outgoing energy flux equals incoming energy flux (in = out). Thus, a Layer Model is exactly in balance.

We will learn about the equations listed here, as well as the Layer Modal, in a little bit. For now, please know that albedo, which is an important component of this model, is defined as a “percentage of incoming radiation reflected back to space”. Thus, albedo is a measure of the reflectivity of the Earth’s surface. Ice, with white snow on top of it, has a high albedo: most sunlight hitting the surface reflects back towards space. Water and brown dirt, for instance, are much more absorbent and less reflective and has a lower albedo than ice.

�if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression� (1896)

ΔF (Wm-2) = α ln(C/C0) α = 5.35 Then, ΔT(K) = λ*ΔF

Svante Arrhenius Climate Sensitivity: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.

IPCC estimate of climate sensitivity 3.8�C � 0.78�C in the SAR 1995 (17 models) 3.5�C � 0.92�C in the TAR 2001 (15 models) 3.26�C � 0.69�C AR4 2007 (18 models)

Range is ~1.5 to 4.5ºC per doubling of CO2 �sensitivity above 4.5 ºC cannot be ruled out�

Arrhenius’s greenhouse law

Svante Arrhenius, a Swedish scientist, received a Nobel Prize for Chemistry in 1903.

Arrhenius developed a theory to explain the ice ages, and he attempted to calculate how

changes in the levels of carbon dioxide in the atmosphere could alter the surface

temperature through the greenhouse effect. Importantly, Arrhenius formulated his

greenhouse law. You can find his greenhouse law, which is the quoted statement in this slide, “if the quantity…”.

His greenhouse law formula remains important at present, and is heavily used. The

greenhouse law formula is:

ΔF = α ln(C/C0) Where C is carbon dioxide (CO2) concentration measured in parts per million by volume

(ppmv); C0 denotes an initial (or reference) CO2 concentration, and ΔF is the change in

the amount of energy reaching the Earth's surface (the radiative forcing) measured in

watts per square meter. Constant α is 5.35. Ln denotes a natural logarithm.

(continue)

�if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression� (1896)

ΔF (Wm-2) = α ln(C/C0) α = 5.35 Then, ΔT(K) = λ*ΔF

Svante Arrhenius Climate Sensitivity: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.

IPCC estimate of climate sensitivity 3.8�C � 0.78�C in the SAR 1995 (17 models) 3.5�C � 0.92�C in the TAR 2001 (15 models) 3.26�C � 0.69�C AR4 2007 (18 models)

Range is ~1.5 to 4.5ºC per doubling of CO2 �sensitivity above 4.5 ºC cannot be ruled out�

Arrhenius’s greenhouse law

The equation indicates that the radiative forcing is proportionate to the increase/decrease in CO2 concentration through time.

Now that we know how to calculate the radiative forcing associated with an increase/decrease in CO2, how do we determine the associated temperature change? In order to project temperature change from the radiative forcing, we need to understand the concept of climate sensitivity.

As described, climate sensitivity is an estimate of how sensitive the climate is to an increase in radiative forcing. The climate sensitivity value tells us how much the planet will warm or cool in response to a given radiative forcing change. Temperature change is proportional to the change in the amount of energy reaching the Earth's surface (the radiative forcing), and the climate sensitivity is the coefficient of proportionality: ΔT = λ*ΔF Where ΔT is the change in the Earth's average surface temperature, λ is the climate sensitivity in Kelvin per Watts per square meter (K/[W/m2]), and ΔF is the radiative forcing. To calculate the change in temperature, we just need to know the climate sensitivity.

ΔF (Wm-2) = α ln(C/C0) α = 5.35 Then, ΔT(K) = λ*ΔF

Svante Arrhenius �Climate Sensitivity�: change in global mean temperature in response to a doubling of CO2 volume mixing ratio.

IPCC (Intergovernmental Panel on Climate Change) estimates climate sensitivity as λ (i.e. how global temperature change responds to changes in climate forcing), and that value (λ) has been modified as we learn more about the variables in our climate system which improves the climate model.

Arrhenius’s greenhouse law

Below is further reading to aid your understand about climate sensitivity: https://www.carbonbrief.org/explainer-how-scientists-estimate-climate-sensitivity

The Greenhouse Effect

σTs 4 =

S 4 1− Albedo( )+σTe

4

σTs 4 = 2σTe

4

σTe 4 =

S 4 1− Albedo( )

Ts = 2 1 4Te

For the surface (1)

For the atmosphere (2)

(1)

(2)

(3)

(4)

Solar flux

Now we are ready to talk about the Layer Model. Suppose we treat the atmosphere as a single layer of gas and that this gas absorbs and re-emits all of the infrared radiation incident on it. Let us assume that it absorbs and emits infrared radiation equally well at all wavelengths, so that we can treat it as a blackbody, and that it has an albedo A in the visible spectrum, just like that of the real Earth.

What are the temperatures of the gas layer and of the surface beneath it? Let’s call the layer temperature Te and the surface temperature Ts.

Let the amount of sunlight striking the planet be equal to S/4 (the globally averaged solar flux) – See figure. The surface absorbs an amount of sunlight striking the planet equal to S/4 x (1-A), along with a flux of downward infrared radiation from the atmosphere equal to sigma*Te4. The atmosphere absorbs an amount of upward infrared radiation from the ground equal to sigma*Ts4, and it emits infrared radiation in both the upward and downward directions at a rate of sigma*Te4 (The real atmosphere also absorbs some of the incoming solar radiation, but we ignore that complication here).

The Greenhouse Effect

σTs 4 =

S 4 1− Albedo( )+σTe

4

σTs 4 = 2σTe

4

σTe 4 =

S 4 1− Albedo( )

Ts = 2 1 4Te

For the surface (1)

For the atmosphere (2)

(1)

(2)

(3)

(4)

Solar flux

Using these energy components, we can write the overall energy balance in the form of two equations.

For the surface (1) and for the atmosphere (2)

The (2) equation arises because the atmosphere radiates in both the upward and downward directions (what goes in, what comes out!). If we now substitute the (2) equation into the left-hand side of the (1) equation and subtract sigma*Te4 from both sides, we obtain (3), which is known as the Earth’s energy-balance formula. Dividing the (3) equation by sigma and then taking the fourth root of both sides yields an additional result (4).

The Greenhouse Effect

σTs 4 =

S 4 1− Albedo( )+σTe

4

σTs 4 = 2σTe

4

σTe 4 =

S 4 1− Albedo( )

Ts = 2 1 4Te

For the surface (1)

For the atmosphere (2)

(1)

(2)

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